Fluid foil



April 1 D. R. DAVI S 2,281,272 v FLUID FOIL Filed May 9, 1938 2 Sheets-Sheet l April 28, 4 D. R. DAvis 2,281,272

' FLUID FOIL Filed May 9, 1558 v 2 Sheets-Sheet 2 Patented pr. 28, 3942 eiieii i 'vi y; eg' iess, Serial n 206,867

" ,ioiaims. (omit- 35) This. invention relates to the oiistruci'ioir of foils. to be driven through a fluid, and-particularly concerns the profiles of the foils. While the invention may be applied to a foil used in any fluid medium, it has its greater-usefulness when applied in the construction of wings for air vehicles.

The upper and lower surfaces of an airfoil should be curved to function inan air stream to H produce lift, without producing excessive drag.

The upper and lower curves of the sectional profile of a fluid foil each affects the fluid flow aroundthe other and they-are, therefore; inter dependenhw, i V

The general object of videfoils having profilesin their. fore-to-aft' sections ,whose upper and lower curves are this invention is-"to pro mutually dependent and'have aparticular relationship, and to provide aiformula havingbut one variable-ico-efficient which if employed in" laying outtheprofile lines for the; upperand lower surfaces of the coils will determines; de-

finite relationship for each I given: type -offoil profiles. i v

A further :object ofthis invention is to" provideua formula, suchuas referred toabove, and

having one variable co-efiicient, 'this variable-coeflicient being constant for the development of thev curve of anygiven foil section, the value of this 'co-efficient determiningthe shape of the 'profile. i

A furtherobject of this inventionis to pro-- videa family of master 'foilsection shapes, detera mined by the formula referred to above, which 35 deviate regularly from each other in profil,*and

in thecharaoter'of their'profile curvature-and which can be readily tested to determinethe most advantageous form of foil section-for" oertain purposes.

A further object of this invention is to provide a familyof master'foilsect'ions, determined by g the formula referred to abova'andto provide a method to vary the thickness of these master sections, without destroying the character of the curves of the upper and lower cambers of said master sections. i

A further object of this invention is tofprovide foil sections anda method'for determining certain profiles forsame, which may be used on a fluid foil to function in a fluid to producea lift curve slopewhich is one hundred percent" efficient, according to the accepted theory;

hereinaiteraf i v v The invention consists 'of'novel parts and com the: Wit-Of h n e ti n w l yappzir bihations of parts to described hereinafter, all of which contribute to produce anmcient I A preferredembodiment ofthe inventionjis described in the followings'pecification, while the broad scope of the 'inventionis pointed out in the appended claims. v y Figure 1 illustrates by rectangular coordinates a basic foil section profile according .to my in=-.

vention.

Figure 2 is a perspective view of a'hal fispan" of an airplane wing constructed in accordance with my 'invention'and tapering looth' in plan and in thickness ratiofIn'this view'the in- 7 crease in thickness ratio "at the root and the dethicknessratio fat the tip-are illuscrease in trated.

Figure l illustratesa'foil section profile em,- bodying-my invention', "and in-whichthe points of the upper and lower profilelines are determined in accordance with" my formula and with reference to a horizontal (X axis) and avertical (Y axis) To develop the 'pa'rticularprofile illus I trated, -co-efi"1ci ent A in "my fomnu'la was given 9 the constant value 1.00. The I mining the profile line i; 2 S

pressed in parametric form:

In order todetermine the profile-line 5,4, whichis expressed employ thefollowing formula, in pammetric form;

.' In all fo i of my equati ll rb i l i formula, values of vtheta are assumed vvarying from zero, to

dis a ws The value of A is constant-forany one foil profile. A sufficient number of values of theta shouldbe; used in order that the numerous enough td'detenmine fair curvea 'By' developing" aseries of master foil sections in which constant A' is given regular increments and'decrements in value, a series of master foil section shapes can be constructed" whichvar'y',"

gradually in their performance'on'account of the fact "that they vary 'very'sli'ghtly from each oth'er,"--"

n s for det ei, V I ofthefoil section are determined for both coordinates in accordtime with the following-"formula, which is ex- Points found will e and the most desirable form may be selected for a given purpose.

In the drawings, chord line I, 6, 4 has been drawn between the foil sections trailing edge I, and leading edge 4, and connecting those two points on the profile which are most widely separate. Line 5, 6, 2 is normal to chord line I, 6, 4 and intersects it at 6, a point 30% of the chord measured rearward from the leading edge 4, or at the 30% station. Hereinafter when the term "chord line is used it shall be defined as a straight line extending throughout the longest distance across a foil section profile and connecting the leading and trailing edge thereof.

In the example illustrated in the drawings, and heretofore described, A being given the value of 1.00, values of theta shown in the following table, when substituted in the formula, result in the values of the restangular coordinates X and Y, shown in the table, which in turn determine the position of the corresponding points on the profile line of the foil section.

tween the upper and lower camber lines and the chord line of the section, at numerous points along the chord. and to express all such distances in percentage of chord length. These percentages are referred to as the foil section ordinates.

For example, the foil section ordinate of the camber line I, 2, 3, 4 at the station along the chord line is equal to 100 times the distance 6, 2 divided by the distance I, 4. And the foil section ordinate of the camber line I, 5, 4 at the 30% station along the chord line is equal to 100 times the distance 6, 5 divided by the distance I, 4.

The foil section ordinates for the upper and/or lower cambers of any master section expressed by my formula may be multiplied or divided by a constant value to produce new ordinates for foil sections having profiles which are proportionally thicker or thinner than the master section from which they were derived, while they retain the characteristics of the master sections profile For example, substitution of the value 0 for 0 in the formulas for X11 and Y results in a value of Xu=0 and a value of Yu=0.726'760. The point I may then be plotted on the XY graph in accordance with these values. Substitution of the value of 1.169370 radians for 0 in the formulas for X and Yu results in a value of Xu=0.251518 and a value of Yu=0.148795. The point satisfying these conditions is the point 2. When a value of 1r/2, 0r 1.570796 is substituted for 0 in the formulas for Xu and Yu, a value of Yu=0 and a value of Xu=0.27324(} result. This point, when plotted, is the point 3. Ordinarily a large number of values of 0, from 0 to 1r/2, are substituted in the formulas for X0 and Yu to de-' termine points between point I and point 3 sufficient to determine accurately the line I, 2, 3. In the same manner, the value 0 may be substituted for 0 in the formulas for X0 and Yb, resulting again in the values Xb=0 and Y1 =-0.726760, again determining the point I. Values of 6:0.430049 and 0:1.028653 may be substituted in the formulas for X0 and Yb, resulting, respectively, in values of Xb=-0.112583, determining the point 5; and Xb=0.234058 Y1 0.137507, determining the point 4. When the value 9=1r/ 2 is substituted in the formulas for, X11 and Yb, the values Xb=0.2'l3240, Yb=0 result, again determining the point 3. Numerous values of '0 may be substituted in the formulas for X11 and Yb to accurately determine the entire profile line from point I around to point 3.

Upon substituting a value 0=1r/2 in the formulae for X and Xb, it will be found that the last term of the formula includes a quantity, (1-20/11'), which equals zero and another quantity, tan 0, which equals infinity. The product of these two quantities is indeterminate. It is, therefore, preferable to evaluate the product (tan 0) (129/7r) as a whole for the value of 0=1r/2. The value of this product is determinate and is 2/1r, so that for 0=1r2 the last term in the formulae for Xu and X0 is 2/11 (1-A).

It is customary in the art to describe a foil section profile by measuring the distances becurvatures in an exaggerated or diminished degree.

This is known in the art as increasing or decreasing the thickness ratio of the foil section. This shown in Figure 2 wherein the dot and dash line designated by the reference numeral I illustrates a basic foil section in accordance with the formulae. The thickness ratio is increased near the root as illustrated by the line designated by the reference numeral 8', the basic foil section according to the formulae being shown by the dot and dash line designated by the reference numeral 8. The thickness ratio is decreased near the tip as illustrated by the dotted line designated by the reference numeral 9', the basic foil section according to the formulae being shown by the dot and dash line designated by the reference numeral 9.

By practicing my invention a series of master foil profiles which deviate gradually from each other in their curvature characteristics, can be produced with mathematical accuracy, and each master profile can be increased or decreased in thickness ratio to produce a series of foil profiles which retain similar contour characteristics, and

this possibility is particularly valuable because when such a series is tested, the performance of any intermediate foil profile can be calculated without test.

In the art, an airfoils efficiency is measured by the value of the airfoils efiiciency factor 1;, which is expressed as follows:

dC /dao (radians) 211' in which C1. is the coefiicient of lift and a0 is the angle of attack corrected for infinite aspect ratio, expressed in radians.

The better foil profiles now known to the art will function on a foil to produce an airfoil efficiency 1 equal to about .9.

Numerous competitive tests in the California Institute of Technology Wind Tunnel on various types of wing models employing the finest profiles now known to the art, on the one hand, and the change in results.

vin which theta varies from- .zero=to" invention function to producean eight per centto twelve per cent increase in airfoil efllciency. One wing model employing sectional profiles derived from the master section-produced by givi'ngxconstant 'A in my formula the value 1.00, the ordinates of'said master profile being increased to give a thickness ratio of 18% at the wingsv root section and being decreasedto a thickness ratio of'12% at the tip section, functioned to'produce an 'efilciency factorf which was greater than 100. According to accepted theory, this wing functioned with an efiiciency of moreithan 100%. Careful check tests were repeated on this airfoil on at least two later occasions, without When functioning at high speed, or a, large Reynolds number, the most eficient master foil section profiles are expressed by my formula when constant A isgiven a value which is not less than .93, or greater than 1.05, and the ordinates for these master profiles may be increased or decreased by a constant per cent to produce allied profile shapes which are particularly efficient. It is to be understood, however, that master profiles derived by using other values of A, and their allied profiles, are useful for particular purposes. Decreasing the value of A results in foils which are thicker relative to the length of the chord line, while increasing the value of A results in foils which are relatively thinner.

The most important portion of a fluid foil sectional profile is the forward 50% thereof. Layout tolerances over the leading half of the foil profiles are comparatively small, while larger tolerances are allowable over the remainder of the foil profile.

It is evident that a fluid foil may utilize one sectional profile at one point in its span, and other section profiles at other points in its span.

The inventor wishes it to be clearly understood that the increased performance enjoyed by fluid foils using his sectional profiles can be directly attributed to the character of the curvature of the upper and lower camber lines, as described by his formula, and that the character of these curves is not changed when the ordinates for a given camber are multiplied or divided by a constant.

This invention accomplishes an important improvement in the art and the discoveries disclosed herein are of great value to all types of aircraft,

throughout their entire flying range.

I claim:

1. A fluid foil having -a sectional profile producible by modification of a sectional profile'jcong forming substantially .to the following 'formula,

the constant.

, and the quantity -A has a value said first profile line conforming substantially to a 2. A fluid foil having a sectional profile producible by modification of a sectional profile conforming substantially to the following formula, in which theta varies from'zero'to T radlans not less than .93 or more than 1.05:

' Xt=sin 0(4/1r1)A+tan 9(120/1r)(1-.A) Yb=COS 0(4/1r+1)A3A (1-20/1r) said modification consisting in multiplying all foil section ordinates of any camber line by a constant value whereby the foil thickness is increased or decreased depending on the value of.

the constant.

3. A fluid foil having a sectional profile defined by a first profile line'and a second profile line,

the following formula:'

in which X and Yu are the rectangular coordious-values from zero-to and the-Lva lue of A isconstant and equal to approximately one:

nates of points on said first profile line determined by holding the value of A constant and equal to approximately one and giving theta various values from zero to and said second profile line conforming substan-' tially to the following formula:

in which Xb and Yb are the rectangular coordinates of points on said second profile line deter- E rad ans 4. A fluid foil having a sectional profile defined by a first-profile line and a second profile' gline, saidffirst profile line conforming substantially to the following formula; Y

natesof points on said first profile line determined .'-,-'by"holdin'g 1A constantat a value not less than .93

andnot greater than 1.05- and givingtheta variv radians and said second-profile line conforming substan- .-tially to the following formula:

in which X): and Ysare therectangular coomp nates of points on said second Iprofile-lineideter- J t ta "various values from jzero Jto DAVID R. DAVIS. 

